Really? You’re Still Counting? (Part 4)

Counting. Nothing in mathematics gives us more of a return on our investment.

In my first blog post in this series, we briefly looked at beginning counters, who start with rote counting and then move on to count concrete items. These are huge investments in their mathematical futures.

In the second post, I discussed and showed examples of what it looks like as students begin to count imagined items. This is the first payoff from all of the counting that they did with concrete items. They can imagine the concrete counting and use that to count “figuratively.”

In the third post, there was some talk and video showing how students begin to count their movements, and then begin to count by just saying number words.

Counting: Counting By Ones – Counting On

At this point, students begin to get huge returns on their investments of counting all by:

Saying a number word sequence

Counting concrete items

Counting in the imagination

Counting movements

Counting number words.

The payoff? Counting on.

With enough experiences of counting from 1 in all those different ways, they begin to “just know” that they don’t have to count from 1 to 8 to know what “eight” means. They’ve done it plenty of times, and they can just start with 8 (first addend) and continue from there.

So if, for instance, a student encounters an additive situation where they need to join 8 and 3, they may say, “8…9, 10, 11” while keeping track of the 3 (second addend) on their fingers or with a spatial or temporal pattern. In the following video, pay close attention to the student’s fingers. You may see her counting to 3 on her fingers but never vocalizing her count. Because of the elapsed time, we know she “counted on” from 8.

Notice: SHE IS STILL COUNTING! Again, we normally call this addition, and that may well be addition. However, she still solves it by counting!

Next time, we will look at what happens as students’ investments of counting on by ones begin to pay off as they find that they can count their counts, leading to success with missing addend, subtractive, and multiplicative situations — STILL BY COUNTING!

About Jason Chamberlain

Jason Chamberlain is a Research Associate for the AIMS Center. His experiences in education include: being a father of two school-age, math-loving children, Holly and Luke; designing off-the-wall math experiences for enthusiastic 3rd grade and less enthusiastic (at least, initially) 8th grade students at Caruthers Elementary School, where he also worked to form a K-8 teacher team for vertical articulation; and facilitating math workshops for the San Joaquin Valley Mathematics Project, California Mathematics Council, and the National Council of Teachers of Mathematics. Jason is a graduate of CSU, Fresno, holds a teaching credential, and anticipates receiving a Master of Arts in Curriculum & Teaching from Fresno Pacific University in 2019. At AIMS, he is currently working to translate research related to the development of the understanding of number in young children.